# Primitive of x by Hyperbolic Cosine of a x

## Theorem

$\displaystyle \int x \cosh a x \rd x = \frac {x \sinh a x} a - \frac {\cosh a x} {a^2} + C$

where $C$ is an arbitrary constant.

## Proof

With a view to expressing the primitive in the form:

$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$

let:

 $\displaystyle u$ $=$ $\displaystyle x$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d u} {\d x}$ $=$ $\displaystyle 1$ Derivative of Identity Function

and let:

 $\displaystyle \frac {\d v} {\d x}$ $=$ $\displaystyle \sinh a x$ $\displaystyle \leadsto \ \$ $\displaystyle v$ $=$ $\displaystyle \frac {\sinh a x} a$ Primitive of $\cosh a x$

Then:

 $\displaystyle \int x \cosh a x \rd x$ $=$ $\displaystyle x \paren {\frac {\sinh a x} a} - \int \paren {\frac {\sinh a x} a} \times 1 \rd x + C$ Integration by Parts $\displaystyle$ $=$ $\displaystyle \frac {x \sinh a x} a - \frac 1 a \int \sinh a x \rd x + C$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \frac {x \sinh a x} a - \frac 1 a \paren {\frac {\cosh a x} a} + C$ Primitive of $\sinh a x$ $\displaystyle$ $=$ $\displaystyle \frac {x \sinh a x} a - \frac {\cosh a x} {a^2} + C$ simplification

$\blacksquare$